Theoretical and experimental study on wave damping inside a rubble mound breakwater

Coastal Engineering 52 (2005) 709 – 725 www.elsevier.com/locate/coastaleng

Theoretical and experimental study on wave damping inside a rubble mound breakwater M.O. Muttray a,*, H. Oumeraci b b

a Delta Marine Consultants b.v., P.O. Box 268, 2800 AG Gouda, The Netherlands Technical University Braunschweig, Leichtweiss Institute, Beethovenstr. 51a, 38106 Braunschweig, Germany

Received 18 December 2003; received in revised form 1 March 2005; accepted 20 May 2005 Available online 1 July 2005

Abstract Wave decay in a rubble mound breakwater has been analysed theoretically for various types of damping functions (linear, quadratic and polynomial). The applicability of these damping functions for wave decay in the landward part of the breakwater core has been investigated in large scale model tests. The properties of the rock materials that have been used in the model tests have been determined to provide a rational basis for the damping coefficients. The analysis is based on detailed measurements of wave conditions and pressure distributions inside the breakwater. The theoretical approaches have been validated and where necessary extended by empirical means. The wave decay inside the breakwater can be reasonably approximated by the commonly applied linear damping model (resulting in exponential wave height attenuation). An extended polynomial approach provides a slightly better fit to the experimental results and reflects more clearly the governing physical processes inside the structure. D 2005 Elsevier B.V. All rights reserved. Keywords: Rubble mound breakwater; Wave damping; Breakwater core

1. Introduction, previous work and methodology The main purpose of rubble mound breakwaters is dissipation of wave energy. However a certain part of the incident wave energy will pass through the break-

* Corresponding author. Tel.: +31 182 590490. E-mail addresses: [email protected] (M.O. Muttray), [email protected] (H. Oumeraci). URLs: http://www.dmc.nl (M.O. Muttray), http://www.xbloc.com (H. Oumeraci). 0378-3839/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2005.05.001

water core resulting in wave disturbance at the lee side of the breakwater. Wave transmission through a breakwater and wave damping inside a breakwater are highly complex processes. The porous flow inside a breakwater is turbulent, non-stationary and non-uniform. In the seaward part of the breakwater, two-phase flow (air–water mixture) is likely to occur. The wave induced pore pressure distribution inside a breakwater is not or only to a minor extent considered in current design procedures. A revision of the actual design procedures has been proposed by

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several authors in the past decades (Bruun and Johannesson, 1976, for example). An improved breakwater design shall include a due consideration of the pore pressure distribution with respect to (i) slope failure analysis, (ii) optimised filter design and (iii) an improved prediction of wave transmission, wave run-up and wave overtopping as well as internal wave set-up (de Groot et al., 1994). A relatively simple analytical method for the assessment of wave height attenuation inside a rubble mound breakwater is presented in this paper. The method has been derived theoretically and validated against experimental results from large scale model tests. The new method reflects the actual physical processes inside the breakwater more than existing approaches and proved to be more accurate. 1.1. Previous work: hydraulic resistance A number of serial and exponential approaches for the hydraulic resistance of coarse porous media have been developed for stationary flow (Hannoura and Barends, 1981). For non-stationary flow in coarse porous media the hydraulic resistance I can be approximated by the extended Forchheimer equation with an additional inertia term (Polubarinova-Kochina, 1962): I ¼ avf þ bjvf jvf þ c

Bvf Bt

b ¼ Kb jo

1n 1 n3 gd

ð3Þ

The coefficient j o is 1 for stationary flow. If coefficient b accounts for viscous and turbulent shear stresses the Forchheimer equation will be applicable not only for combined laminar–turbulent flow, but also for fully turbulent flow (van Gent, 1992a). The hydraulic resistance of a uniform non-stationary flow is described by the extended Forchheimer equation (Eq. (1)). For oscillatory flow the additional resistance with regard to the convective acceleration has to be considered by a quadratic resistance term (Burcharth

ð1Þ

where a, b and c are dimensional coefficients and v f is the flow velocity inside the porous medium (filter velocity). A set of widely used theoretical formulae for the Forchheimer coefficients a, b and c (see van Gent, 1992a) is presented in this section. Alternative approaches can be found in literature. Especially for coarse and wide graded rock material it might be necessary to determine the resistance coefficients experimentally. Kozeny (1927) derived the coefficient a for stationary flow by considering the porous flow as capillary flow and the porous medium as a matrix of spherical particles of equal size: ð 1  nÞ 2 m a ¼ Ka gd 2 n3

man (1937), Ergun and Orning (1949), Ergun (1952), Koenders (1985), den Adel (1987), Shih (1990), van Gent (1992a) and by Burcharth and Andersen (1995). Various coefficients K a that have been derived by these authors are listed in Table 1. The large variability of the coefficient K a should be noted. The following approach for the resistance coefficient b, which includes the non-dimensional coefficient K b (see Table 1), has been proposed by Ergun and Orning (1949), Ergun (1952), Engelund (1953), Shih (1990), van Gent (1992a) and Burcharth and Andersen (1995) for stationary flow:

Table 1 Empirical coefficients and characteristic particle diameters for the Forchheimer coefficients a and b (stationary flow) Author

Characteristic particle diameter

Dimensionless Ka

Coefficients Kb

Kozeny (1927), Carman (1937) Ergun (1952), Ergun and Orning (1949) Engelund (1953) Koenders (1985) den Adel (1987) Shih (1990) van Gent (1993)

d eqa

180

–c

d n50b

150

1.75

d eq d n15 d n15 d n15 d n50

–c 250–330 75–350 N1684 1000

1.8–3.6 –c –c 1.72–3.29 1.1

a

ð2Þ

with porosity n, grain size d, non-dimensional coefficient K a and kinematic viscosity m. Eq. (2) has been confirmed theoretically and experimentally by Car-

The equivalent diameter d eq is the diameter of a sphere with mass m 50 and specific density . s of an average actual particle: d eq = (6m 50 / k. s )1 / 3 . b The nominal diameter d n50 is the diameter of a cube with mass m 50 and specific density . s of an average actual particle. d n50 = (m 50 / . s )1 / 3 . c Authors proposed a different approach from Eqs. (2) and (3).

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and Andersen, 1995). Hence, the resistance coefficient b will be increased. van Gent (1993) determined experimentally a coefficient j o of 1 + 7.5 / KC for oscillatory flow. The Keulegan–Carpenter number KC = v˜ fT / (nd) characterises the flow pattern (with velocity amplitude v˜ f and period T) and the porous medium (particle size n and diameter d). The inertia coefficient c will not be affected by the convective acceleration and reads:
 1 1n 1 þ KM c¼ ð4Þ ng n This approach has been used by Sollit and Cross (1972), Hannoura and McCorquodale (1985), Gu and Wang (1991) and by van Gent (1992a). The added mass coefficient K M will be 0.5 for potential flow around an isolated sphere and for a cylinder it will be 1.0. In a densely packed porous medium the coefficient K M cannot be determined theoretically; most probably it will tend to zero (Madsen, 1974). van Gent (1993) proposed the following empirical equation for the added mass coefficient of a rubble mound. His approach may lead for small velocity amplitudes v˜ f and long periods T to negative values of K M, which are physically meaningless and have to be excluded:   ngT ;0 KM ¼ max: 0:85  0:015 v˜ f The hydraulic resistance of a rigid homogeneous, isotropic porous medium can be determined for single phase flow by Eqs. (1), (2), (3) and (4). Using the Forchheimer coefficients an averaged Navier Stokes equation (van Gent, 1992a) reads:
 BvYf 1 Y p Y c þz þ 2 vf dgrad vf ¼  grad ng .g Bt  avYf  bjvYf jvYf

ð5Þ

Neglecting the convective acceleration yields the extended Forchheimer equation (Eq. (1)). In case of a non-rigid porous medium the combined motion of fluid and particles has to be considered as two-phase flow. For an anisotropic porous medium the Forchheimer coefficients may vary with flow direction. A spatial variation of Forchheimer coefficients has to be considered for inhomogeneous porous media. The transition between areas with different resistance has to

711

be defined separately. Air entrainment leads to a compressible two-phase flow. A modified Forchheimer equation for two-phase flow (air–water mixture) in a porous medium has been proposed by (Hannoura and McCorquodale, 1985). 1.2. Previous work: pore pressure oscillations in rubble mound breakwaters The wave propagation in a rubble mound breakwater has been investigated by Hall (1991, 1994) in small scale experiments and by Buerger et al. (1988), Oumeraci and Partenscky (1990) and Muttray et al. (1992, 1995) in large scale experiments. Field measurements have been conducted at the breakwater at Zeebrugge (Troch et al., 1996, 1998), prototype data, experimental data and numerical results have been analysed by Troch et al. (2002). The water surface elevations inside the breakwater and the amplitude of the pore pressure oscillations decrease in direction of wave propagation exponentially (Hall, 1991; Muttray et al., 1995). However, for larger waves Hall (1991) stated constant maximum water surface elevations in the seaward part of the breakwater core, which might be induced by dinternal wave overtoppingT (the wave run-up on the breakwater core reaches the top of the core and causes a downward infiltration from the crest). The water surface elevations, the pore pressure oscillations and the wave setup increase with increasing wave height, wave period and slope cot a (Oumeraci and Partenscky, 1990; Hall, 1991). They decrease with increasing permeability of the core material and with increasing thickness of the filter layer (Hall, 1991). The damping rate of pore pressure oscillations increases with wave steepness (Buerger et al., 1988; Troch et al., 1996) and decreases with increasing distance from the still water line (Oumeraci and Partenscky, 1990; Troch et al., 1996). The following approach has been proposed by Oumeraci and Partenscky (1990) and applied by Burcharth et al. (1999) and Troch et al. (2002) for the damping of pore pressure oscillations:
 2p Pð xÞ ¼ P0 exp  Kd x ð6Þ LV with dimensionless damping coefficient K d, amplitude of pore pressure oscillations P 0 (at position x = 0) and

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P(x) (at varying position x N 0) and wavelength LV inside the structure. The exponential decrease has been confirmed in field measurements (Troch et al., 1996). Numerical models have been developed to study the pore pressure attenuation inside rubble mound breakwaters. van Gent (1992b) proposed a one-dimensional model where the hydraulic resistance of the porous medium has been considered by Forchheimer type resistance terms. This model provides a realistic figure of the internal wave damping for a wide range of wave conditions and structures. Troch (2001) developed a numerical wave flume in order to study the pore pressure attenuation inside a rubble mound breakwater (2-dimensional analysis). An incompressible fluid with uniform density has been modelled; the VOF method has been applied to treat the free surface. The Navier Stokes equation has been extended by Forchheimer type resistance terms for the porous flow model. However, the effect of air entrainments into the breakwater core has not been considered. From comparison of numerical results, prototype measurements and experimental results it has been concluded that the wave damping inside a breakwater can be approximated by an exponential function (Troch, 2001; Troch et al., 2002). 1.3. Methodology This study is based on a theoretical analysis of the wave damping inside a rubble mound breakwater. A simple universal concept for the wave height decay inside the structure is presented. Large scale experiments have been performed with a rubble mound breakwater of typical cross section in order to validate the theoretical results. A large model scale has been selected to prevent scale effects especially with respect to air entrainment. The hydraulic model tests were intended to provide insight into the physical processes of the wave structure interaction, to confirm the theoretical damping approach and to quantify non-linear effects that had been neglected in the theoretical approach. The final objective is a simple and relatively accurate description of wave damping inside a rubble mound breakwater, which clearly reflects the governing physical processes.

It should be noted that the oscillations of water surface (wave height) and pore pressure (dpore pressure heightT) are closely linked. The wave kinematics in porous media vary significantly in direction of wave propagation. However, the local wave kinematics (at a specific location) are similar inside and outside the porous medium. A linear relation between the wave height at a specific location and the corresponding height of the pore pressure oscillations at a certain level below SWL has been derived theoretically by Biesel (1950). Thus, a proper description of the wave damping inside a breakwater should be applicable for wave heights and pore pressure oscillations. The analysis of the wave propagation inside the breakwater core has been restricted to the breakwater part, where the water surface remains inside the core during the entire wave cycle. Thus, only the breakwater core landward of the point of maximum wave run-up between the filter layer and the core has been considered (Fig. 1). Further seaward the wave motion inside the breakwater core will be affected by the various layers of the breakwater (with varying permeability) and by the wave motion on the breakwater slope (wave run-up and run-down). The wave height at the point of maximum wave run-up between the core and the filter layer (x 0 = x(R u, max)) has been considered as the initial wave height H 0. This procedure is different from previous studies, where the interface between filter layer and core has been considered as x 0 (see for example Buerger et al., 1988; Troch et al., 1996; Burcharth et al., 1999). With the proposed definition of the position x 0, the wave damping will not be affected by the breakwater geometry but only by the hydraulic resistance of the core material. The wave transformation in the seaward part of the breakwater and the relation between incident wave height, wave run-up and initial wave height H 0 are not subject of this paper. A homogeneous core material has been assumed for the wave damping analysis. It has been further assumed that the wave damping inside the core at x z x 0 is not directly affected by the wave transformation at the seaward slope (except for the initial wave height H 0). Thus, wave damping inside the breakwater core and the infiltration process at the slope have been considered as two separate and

M.O. Muttray, H. Oumeraci / Coastal Engineering 52 (2005) 709–725

2

713

z

elevation z [m]

ηmax(x) 1

H0

R u,C R C

SWL

0

x

x0 ηmin(x)

-1

247

249

regular waves: h = 2.49m; T = 8.0s; H = 1.00m 251

253

255

distance to wave generator [m] Fig. 1. Definition sketch for the analysis of wave damping inside the breakwater core.

successive processes. The following terminology is used in this paper: Water surface oscillation and pressure oscillation stand for the variation of water surface line and dynamic pressure in time (at a specific location x). Wave height H refers to the vertical distance between wave crest and trough at a specific location x for regular waves and to the significant wave height H m0 for irregular waves. Pressure height P corresponds to wave height (derived from pressure oscillations instead of water surface elevations). Wave damping denotes the attenuation of wave height and pore pressure height in direction of wave propagation (x-direction). Deviations between observation and theoretical prediction of a parameter y are quantified by standard deviation r y, relative standard deviation r y / y¯ (with mean value y¯ ) and by systematic deviation l y (average deviation between observation and prediction).

2. Theoretical approach for wave damping The relationship between hydraulic gradient I and discharge velocity v f is determined by the extended Forchheimer equation (Eq. (1)). The wave damping inside a rubble mound breakwater is closely linked to the hydraulic resistance. Various relatively simple approaches for the wave damping inside a rubble mound breakwater have been derived from Eq. (1). The discharge velocity v f (x, z, t) has been replaced by a depth averaged (z = 0 Y  h) and time averaged

(t = 0 Y T) velocity v¯f (x). The averaged particle velocities inside the breakwater v¯ f (x) can be approximated according to Muttray (2000) by: v¯ f ð xÞ ¼ jv H ð xÞ

ð7Þ


 n x 2 coshkVh 1þ 1 jv ¼ p kVh p cosh1:5kVh

ð8Þ

with local wave height H(x), circular frequency x = 2k / T, internal wave number kV = 2k / LV (internal wavelength LV), water depth h and porosity n. The velocity coefficient j v[s 1] has been introduced for convenience only. The hydraulic gradient I¯ (x) (averaged over water depth and wave period) corresponds to the averaged pressure gradient if the water depth is constant. The gradient I¯ (x) will be equal to the gradient of the pressure height BP¯ (x) = Bx and to the gradient of the wave height BH(x) = Bx if the oscillations of water surface and pore pressure (variation in time at a fixed location) are sinusoidal and if possible variations of pore pressure oscillations over the water depth are neglected:
 ¯I ð xÞ ¼  grad p¯ ð xÞ .g 2 BP¯ ð xÞ 2 BH ð xÞ ¼  ¼  k Bx k Bx

ð9Þ

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With hydraulic resistance ¯I (x) according to Eq. (9) and discharge velocity v¯ f according to Eq. (7) the Forchheimer equation (with B(j vH) / Bt = 0) reads: 2 BH ð xÞ ¼ ajv H ð xÞ þ bð jv H ð xÞÞ2  k Bx

can be determined from the boundary conditions if the initial wave height is H(x = 0) = H 0: C ¼ lnðH0 Þ

ð10Þ

The resulting damping function will be linear, quadratic or polynomial and depends on the actual flow properties.

and the wave height decay due to linear damping is finally described by:  k  ð11Þ H ð xÞ ¼ H0 exp  ajv x 2

2.1. Linear damping

2.2. Quadratic damping

For laminar flow, the relation between hydraulic gradient and discharge velocity is linear (Darcy’s law). A linear dependency between wave height decay and wave height is called linear damping. If non-linear hydraulic resistance is neglected Eq. (10) reduces to:

The hydraulic gradient in fully turbulent flow is in proportion to the discharge velocity squared. A wave height decay that depends on the wave height squared is called quadratic damping. If the linear hydraulic resistance is neglected Eq. (10) leads to the following quadratic damping function:



2 BH ð xÞ ¼ ajv H ð xÞ k Bx

2 BH ð xÞ ¼ bð jv H ð xÞÞ2 k Bx 1 1 H ð xÞ ¼ k with : C ¼  2 H 0 bj x  C 2 v H0 H ð xÞ ¼ p 2 bj H0 x þ 1 2 v 

Separation of variables and integration leads to:
 2 H ð xÞ ¼ exp  ajv x þ C k A linear damping thus results in an exponential decrease of wave height. The integration constant C

1 type of damping function: linear: δH(x)/δx = -1 H(x) quadratic: δH(x)/δx = -1 H(x) 2 polynomial: δH(x)/δx = -1 H(x) 1.5

wave height H [m]

0.8

0.6

quadratic

0.4

0.2 linear 0 polynomial 0

1

2

3

position x [m] Fig. 2. Linear, quadratic and polynomial damping.

4

5

ð12Þ

M.O. Muttray, H. Oumeraci / Coastal Engineering 52 (2005) 709–725

715

Table 2 Hydraulic resistance coefficients and initial wave heightsa Damping-function

Coefficient a[s/m]

Coefficient b[s2/m2]

Initial wave height H 0[m]

Linear Quadratic Polynomial

2 / (kj v) – 1 / (kj v)

– 2 / (kj 2v) 1 / (kj 2v)

0.5–1.0 0.5–1.0 0.5–1.0

a

Applied in Fig. 2.

2.3. Polynomial damping

Two waves with initial wave heights H 0 (at position x = 0) of 1 and 0.5 m are considered (see Fig. 2). The ratio of the local wave heights H(x) (at position x N 0) will be constant (= 0.5) for linear damping. Quadratic damping will cause a stronger wave height reduction for the larger wave and consequently, the ratio of the local wave heights will vary (from 0.5 to 1). A similar effect can be seen for polynomial damping, which of course depends on the relative importance of the quadratic resistance.

The hydraulic gradient can be described by the Forchheimer equation if turbulent and laminar flow both occur in the porous medium. A wave height decay that depends on the wave height and on the wave height squared is called polynomial damping. Eq. (10) leads to the following polynomial damping function: 2 BH ð xÞ ¼ ajv H ð xÞ þ bð jv H ð xÞÞ2 k Bx kajv hk i H ð xÞ ¼ with : 2exp ajv ð x þ C Þ  kbj2v 2
 2 k a C¼ ln jv þ bjv kajv 2 H0 a   H ð xÞ ¼
ð13Þ  a k þ bjv exp ajv x  bjv H0 2 

3. Experimental investigations 3.1. Experimental set-up and test procedure The experimental set-up in the Large Wave Flume (GWK) in Hanover consisted of a foreshore (length 100 m, 1 : 50 slope) and a rubble mound breakwater of typical cross section with Accropode armour layer, underlayer, core, toe protection and crest wall. The

The various damping functions are plotted in Fig. 2; the coefficients a and b and the initial wave heights H 0 that have been used are listed in Table 2.

wave gauges & wave run-up gauges 17

18

19

20

21

22

23

24

1

25

2

+4.30m

26

+3.75m

34

SWL

+2.90m +2.50m

33 31

+1.50m +0.60m ±0.00m

2.00m

240

29

30

25

24

19 1

32

27

28 23 22 14

27

3

15

16

17

10

11

12

5

6

18

26

21 20 8

9

13

pressure cells 2

3

4

7

flume bottom

244

248

252

256

260

distance to wave generator x [m] Fig. 3. Cross section of the breakwater model with measuring devices: wave gauges, wave run-up gauges and pressure cells.

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breakwater had 1 : 1.5 slopes and the crest level was at 4:50 m above seabed (Fig. 3). The breakwater core had a crest width of 1.35 m and a crest height of 3.75 m (Fig. 3). The core material consisted of gravel (rock size 22/56 mm); the geometric properties of the core material are summarised in Table 3. The resistance coefficients of the extended Forchheimer equation for oscillating single phase flow inside the breakwater core are summarised in Table 4 as well as the their contribution to the total flow resistance. The coefficients for the core material have been determined experimentally for stationary and oscillatory flow conditions that were similar to the tested flow conditions inside the breakwater. The wave motion on the breakwater slope, which is governing the hydraulic processes inside the structure, was measured including wave run-up, pressure distribution on the slope and water surface elevations. The wave propagation inside the structure has been determined from the water surface elevations inside the core (wave gauges 22–26) and at the boundaries between different layers (wave run-up gauges 2 and 3). The pore pressure distribution was measured inside the core (at three different levels: pressure cells 1–7, 8–13, 14–18) and along the boundaries of the various layers (pressure cells 24–28 and 19–23). The positions of the measuring devices at and inside the rubble mound breakwater are specified in Fig. 3. For wave height measurements inside the structure the wave gauges were protected by a cage against the surrounding rock material. Pressure sensors of type Druck PDCR 830 were used. For pore pressure measurements, the pressure cells were protected by a plastic shell. A typical breakwater configuration has been selected for the model tests in the GWK; the structural parameters (geometry and rock material) have not been varied with respect to the large model scale. The wave parameters have been varied systematically (see Table 5). Table 3 Geometric properties of core material (van Gent, 1993) Equivalent diameter Nominal diameter

Non-uniformity Porosity

d eq = 0.0385 m d n15 = 0.023 m d n50 = 0.031 m d n85 = 0.040 m d n60 / d n10 = 1.51 n = 0.388

d eq = (6m 50 / k. s )1 / 3 d ni = (m i / . s )1 / 3

Table 4 Resistance coefficients for the core material and contribution to the total flow resistance for oscillatory flow Laminar Resistance Turbulent Resistance Inertia Force coeff. a contribution coeff. b contribution coeff. contribution [s/m] [%] [s2/m2] [%] c[s2/m] [%] 0.89

11

22.9

83–87

0.26

2–6

Two water levels (h = 2.50 and 2.90 m) have been tested. At the lower water level (h = 2.50 m) most of pressure cells were permanently submerged, thus providing a complete picture of the pore pressure oscillations. Wave overtopping was practically excluded from these tests in order to avoid any infiltration into the breakwater core from the breakwater crest. Therefore the wave heights at water level h = 2.50 and 2.90 m were limited to H = 1.00 and 0.70 m, respectively. Tests were conducted with both regular and irregular waves. The regular wave tests were used to provide insight into the hydraulic processes, but also to validate the theoretical approaches and to develop empirical adjustments and extensions. Tests with irregular waves (TMA-wave spectra generated from JONSWAP-spectra) were used to check the applicability of regular wave results for irregular waves and if necessary to adapt it. The wave conditions that have been tested are summarised in Table 5. The relative water depth h / L varied from 0.05 to 0.23 (kh = 0.35–1.45) and was thus transitional and close to shallow water conditions. The wave steepness (H / L = 0.005–0.056) was significantly lower than the limiting wave steepness for progressive waves ((H / L)crit c 0.14) and also the relative wave height (H / h = 0.09–0.4) was significantly lower than the critical wave height ((H / L) c 0.8). The surf ffiffiffiffiffiffiffiffiffiffiffi pcrit similarity parameter n ¼ tana= H=L0 varied from 3.0 to 16.7. The ratio of initial and incident wave height H 0 / H i was 0.35–1.14 (on average 0.68). Typical results of the hydraulic model tests are plotted in Fig. 4 showing the water surface line and the pressure distribution inside the breakwater just before maximum wave run-up. 3.2. Experimental results The wave height inside the breakwater core is plotted in Fig. 5 against the distance x  x 0 (see Fig. 1) for regular waves (wave periods T = 4, 8 s,

M.O. Muttray, H. Oumeraci / Coastal Engineering 52 (2005) 709–725

717

Table 5 Wave conditions tested Water level h [m]

Wave period T, T p

Regular waves Incident wave heighta H i [m]

Initial wave heightb H 0 [m]

Incident wave heighta H i, m0 [m]

Initial wave heightb H 0, m0 [m]

2.50 2.50 2.50 2.50 2.50 2.90 2.90 2.90 2.90 2.90 2.90

4 5 6 8 10 3 4 5 6 8 10

0.23–0.81 0.26–1.06 0.27–1.09 0.28–1.25 0.60 0.22–0.60 0.23–0.70 0.25–0.70 0.26–0.73 0.27–0.78 0.27–0.81

0.15–0.78 0.19–0.98 0.23–1.04 0.28–1.06 0.59 0.11–0.21 0.14–0.31 0.18–0.38 0.22–0.46 0.27–0.61 0.31–0.73

0.24–0.78 0.25–0.98 0.25–1.04 0.26–1.06

0.15–0.34 0.19–0.49 0.21–0.60 0.27–0.77

0.22–0.47 0.23–0.64 0.24–0.68 0.26–0.71 0.26–0.73 0.27–0.58

0.11–0.18 0.14–0.29 0.18–0.37 0.22–0.45 0.26–0.58 0.31–0.57

a b

Wave spectra

At the toe of the breakwater. Inside the breakwater core.

nominal incident wave heights (in front of the breakwater) H = 0.25, 0.40, 0.55, 0.70 m and water level h = 2.50 m). The initial wave height H 0 at position x 0 varies with incident wave height and with wave period. For shorter waves (T = 4 s) the wave height inside the breakwater core H(x) approaches after a relatively short distance a value that is almost independent of the initial wave height H 0. Once this value is reached the wave damping is significantly reduced. A similar effect (decreasing variation of local wave heights and decreasing wave damping) can also be seen for longer waves (T = 8 s). The varying ratio of the local wave heights (see Figs. 2 and 5) gives some indication that the wave

elevation z [m]

2 1

3.2.1. Linear damping The linear damping approach (Eq. (11)) uses a resistance coefficient a, which corresponds to the

test: 200694-01 h = 2.52m; T = 8s; Hi = 1.01m t = t(Ru,max) + 7.50s

pressure cell 0.7 0.8

0.9

1.0 η(t)

0

0.2

-1 0.6

-2

damping inside a breakwater will be better approximated by a quadratic or polynomial damping function than by the linear damping function. The applicability of the damping functions (Eqs. (11) (12) and (13)) for the wave height attenuation inside a breakwater is addressed below. The velocity coefficient j v according to Eq. (8) was almost constant r j v c 0.25 s 1 with a standard deviation r j v = 0.011 s 1 (4.5%)) for the wave conditions that have been tested (see Table 5).

0.5

0.1

0.4 0.3

0.5

247

249

251

253

255

257

distance from wave generator X [m] Fig. 4. Lines of constant pressure heights p / .g [m] inside the breakwater during wave run-up (regular wave, T = 8 s, H i = 1.01 m, h = 2.52 m).

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M.O. Muttray, H. Oumeraci / Coastal Engineering 52 (2005) 709–725

0.75 wave parameters: h = 2.50m; H = 0.25 - 0.70m T = 4s T = 8s

wave height H [m]

0.6 0.45 0.3 0.15 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

distance x - x0 [m] Fig. 5. Variation of wave height inside the core with distance x–x 0 for various incident wave heights and wave periods.

Forchheimer coefficient a for laminar flow. However, for turbulent flow the linear Forchheimer coefficient will not be applicable. As the flow pattern inside a breakwater is governed by turbulent flow the coefficient a has been replaced by a coefficient a eq that takes laminar and turbulent resistance into account. The coefficient a eq has been derived by a linearisation, i.e. integration and averaging of the actual flow resistance. A rational approximation for the linearised resistance coefficient has been proposed by Muttray (2000):   1 gb kVh 2 aeq ¼ a þ H¯ pffiffiffiffiffi 1 þ  tanh kVh ð14Þ 36 4 5 ch ¯ inside the breakwater with average wave height H core and internal wave number kV, which has been assessed from a linearised dispersion equation for wave motion in porous media: x2 ¼

kV tanhðkVhÞ nc

ð15Þ

The coefficient a eq has been determined iteratively. A starting value for a eq has been derived from Eq. ¯ = H 0 (see Table 5) and coefficients a, b (14) with H and c according to Table 4. The wave height inside the breakwater H(x) has been determined subsequently ¯ (averaged from Eq. (11). The average wave height H over the width of the breakwater) has finally been used to recalculate a eq from Eq. (14). The width of the breakwater core that has been tested varied at SWL between 3.9 and 5.1 m (depend-

ing on the water level); a distance of 3 m has been considered for the assessment of the average wave ¯ . For the wave conditions tested the average height H ¯ = 0.43 H 0. The internal wave height was about H resistance coefficient a eq varied between 1.2 and 3.6 and was on average 2 with standard deviation r a = 0.55 s/m (27%). The applicability of the linear damping approach (Eq. (11)) for the decrease of wave height inside the breakwater core is demonstrated in Fig. 6. In Fig. 6a the coefficient a eq that has been derived from Eq. (11) (with measured wave heights H 0 and H(x)) is plotted against the relative water depth kVh. The coefficient a eq is on average 2 s/m; the standard deviation r a is 0.803 s/m (38.8%). The theoretical assessment of the linearised resistance coefficient a eq (iterative procedure) is confirmed by these results. The differences between measured and calculated wave height decay (according to Eq. (11)) are shown for a number of tests with shorter and longer wave periods (T = 4 s and T = 8 s, H i = 0.25  0.70 m and h = 2.90 m) in Fig. 6b (with a eq = 2 s/m and j v c 0.25 s 1). The wave height attenuation of longer waves (T = 8 s) is overestimated in the seaward part of the breakwater core (j vx V 0.3) while the wave damping of shorter waves (T = 4 s) is underestimated. The wave height evolution inside the core is plotted in Fig. 6c against the relative distance j vx for all tests. The non-dimensional distance j vx is a governing parameter for the wave attenuation according to the linear damping approach (Eq. (11)). It can be seen

M.O. Muttray, H. Oumeraci / Coastal Engineering 52 (2005) 709–725

(a) Coefficient a eq

(b) Wave Damping: T = 4s and T = 8s

σa = 0.803s/m (38.8%)

1

6

H(x)/H0 [-]

coefficient aeq [s/m]

8

4 2

wave parameters: H = 0.25 - 0.70m h = 2.90m H(x)/H 0 = exp(-π/2 κ v a eq x) T = 4s T = 8s

0.8 0.6 0.4 0.2

aeq = 2 0

0 0

0.4

0.8

1.2

1.6

2

0

relative water depth k’h [-]

(c) Damping Function

0.8 0.6

0.2

0.4

0.6

0.8

relative distance κv x [m/s]

1

(d) Measurement & Prediction 0.8

T = 3s T = 4s T = 5s T = 6s T = 8s T = 10s

calculated H(x) [m]

H(x)/H0 = exp(-π/2 κv aeq x)

1

H(x)/H0 [-]

719

0.4 0.2 0

σH = 0.028m (18.2%) µH = 0.975

0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

relative distance κv x [m/s]

1

0

0.2

0.4

0.6

0.8

measured H(x) [m]

Fig. 6. Linear wave damping approach: (a) damping coefficient a derived from measurements, (b) wave decay for longer (T = 8 s) and shorter (T = 4 s) wave periods, (c) wave decay vs. relative distance j vx and (d) comparison of measured and calculated local wave heights H(x).

that the wave heights in the landward part of the breakwater core (j vx V 0.6) are underestimated. A direct comparison of measured and calculated wave heights inside the structure is plotted in Fig. 6d for all tests. The measured wave heights are on average 2.5% smaller than the calculated wave heights; the standard deviation is r H = 0.028 m (18.2%). Even though the actual wave height decay inside the structure shows some systematic differences as compared to the calculated decay according to Eq. (11), the linear damping approach provides a very simple and relatively accurate approximation for the wave damping inside the breakwater core (see Fig. 6c, d). 3.2.2. Quadratic damping The applicability of the quadratic damping approach according to Eq. (12) for the wave height decay inside the breakwater core is shown in Fig. 7. The quadratic damping coefficient b that has been derived from Eq. (12) (with measured wave heights H 0 and H(x)) is plotted in Fig. 7a against the relative water depth kVh. The coefficient b is increasing with

kVh and has been approximated empirically by Eq. (3) and j oK b = 3.4 kVh. The standard deviation between observed and calculated coefficients b is r b = 42.4 s2/ m2 (86.1%). The Forchheimer coefficient b = 22.9 s2/ m2 that has been determined experimentally (see Table 4) is plotted for comparison. The differences between measured and calculated wave height decay (according to Eq. (12)) are shown for a number of tests with shorter and longer wave periods in Fig. 7b (see also Fig. 6b). The decrease of wave height is plotted against j v2H 0x. The damping coefficient b varies with kVh; the wave damping is therefore different for longer and shorter waves. The wave damping inside the breakwater is plotted in Fig. 7c against the relative distance bj v2H 0x for all tests. The non-dimensional distance bj v2H 0x is a governing parameter of the quadratic damping approach (Eq. (12)). The wave heights in the seaward part of the structure (bj v2H 0x V 0.5) are slightly underestimated while the wave heights in the landward part (bj v2H 0x N 1) are slightly overestimated.

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M.O. Muttray, H. Oumeraci / Coastal Engineering 52 (2005) 709–725

(a) Coefficient b

(b) Wave Damping: T = 4s and T = 8s 1.2

σb = 42.4s2/m2 (86.1%)

144

H* = H(x)/H0 [-]

coefficient b [s 2 /m2 ]

180

108 72

Kb κ0 = 3.4 k’h b = 22.9 s2/m2

36 0

wave parameters: H = 0.25 - 0.70m h = 2.90m

1 0.8

H* = (12 π κv2 H0 x + 1)-1

0.6

H* = (36 π κv2 H0 x + 1)-1

0.4 0.2 0

0

0.3

0.6

0.9

1.2

1.5

1.8

0

relative water depth k’h [-]

(c) Damping Function

0.8 0.6 0.4

H* =

(0.5 π b

κ v2 H 0 x

+

calculated H(x) [m]

T = 3s T = 4s T = 5s T = 6s T = 8s T = 10s

1

0.02

0.04

0.06

0.08

relative distance κv 2 H0 x [m2 /s2 ]

0.1

(d) Measurement & Prediction 0.8

H* = H(x)/H0 [-]

T = 4s T = 8s

1)-1

0.2 0

σH = 0.032m (21.1%) µH = 0.999

0.6 0.4 0.2 0

0

1

2

3

4

relative distance b κv 2 H0 x [-]

5

0

0.2

0.4

0.6

0.8

measured H(x) [m]

Fig. 7. Quadratic wave damping approach: (a) damping coefficient b derived from measurements, (b) wave decay for longer (T = 8 s) and shorter (T = 4 s) wave periods, (c) wave decay vs. relative distance bj 2vH 0x and (d) comparison of measured and calculated local wave heights H(x).

A direct comparison of measured and predicted wave heights (Eq. (12)) can be seen in Fig. 7d for all tests. The systematic deviation between measured and calculated values is only 0.1%, the standard deviation is r H = 0.032 m (21.1%). Major differences can be seen for larger wave heights (H(x) N 0.3 m), therefore the goodness of fit is slightly less than for the linear damping approach. If the Forchheimer coefficient b = 22.9 s2/m2 (see Table 4) is applied as a damping coefficient in Eq. (12) the wave height inside the structure will be overestimated on average by 16.3%. The standard deviation between measured and predicted wave heights will be increased to r H = 0.052 m (33.9%). The wave decay inside the breakwater is qualitatively better reproduced by the quadratic damping function than by the linear damping model (see Figs. 6 and 7b). However, the effect of wavelength is apparently not properly described by the velocity coefficient j v. This shortcoming will affect the results from the quadratic approach more than the results of the linear approach (j v is linear in Eq. (11) and

quadratic in Eq. (12)). Even though the quadratic approach covers more of the physics of the wave damping process inside a breakwater than the linear approach it does not provide a better prediction of the wave height attenuation. The shortcomings of the velocity coefficient j v can be partly compensated by an empirical correction of coefficient b (with j oK b = 3.4 kVh). For the present case the relative standard deviation has been reduced from 33.9% to 21.1%. However, the general applicability of such purely empirical procedure is uncertain. In the landward part of the breakwater the wave heights are underestimated by the linear damping approach and overestimated by the quadratic approach. Thus, a polynomial approach according to Eq. (13) that contains a linear and quadratic contribution might be a better alternative. 3.2.3. Polynomial damping The applicability of the polynomial damping approach according to Eq. (13) is shown in Fig. 8.

M.O. Muttray, H. Oumeraci / Coastal Engineering 52 (2005) 709–725

(a) Coefficient a

(b) Coefficient b 150

σa = 0.193s/m (23.0%) K b κ 0 = k’h

1.5

coefficient b [s2 /m2 ]

coefficient a [s/m]

2

a = 0.89 1 0.5

σb = 19.9s2/m2 (93.0%) a = 0.89s/m

120 90

Kb κ0 = k’h

60 30

b = 22.9 s2/m2

0

0 0

0.4

0.8

1.2

1.6

2

0

relative water depth k’h [-]

H(x)/H0 [-]

0.4

calculated H(x) [m]

T = 3s T = 4s T = 5s T = 6s T = 8s T = 10s

0.6

0.8

1.2

1.6

2

(d) Measurement & Prediction 0.8

0.8

0.4

relative water depth k’h [-]

(c) Damping function 1

721

damping function

0.2 0

σH = 0.026m (16.8%) µH = 1.022

0.6 0.4 0.2 0

0

2

4

6

8

10

(a + b κv H0 ) exp(π/2 a κ v x) - b κ v H0

0

0.2

0.4

0.6

0.8

measured H(x) [m]

Fig. 8. Polynomial wave damping approach: (a) linear damping coefficient a derived from measurements, (b) quadratic damping coefficient b derived from measurements, (c) wave decay inside the breakwater core and (d) comparison of measured and calculated local wave heights H(x).

The linear and quadratic resistance coefficients a and b are plotted in Fig. 8a and b against the relative water depth kVh. The Forchheimer coefficient a = 0.89 s/m (see Table 4) has been applied as linear damping coefficient. The standard deviation between coefficients a that have been deduced from measurements (with empirically corrected coefficient b) and the nominal value a = 0.89 s/m is r a = 0.193 s/m (23.0%). The quadratic damping coefficient b has been corrected empirically in order to compensate the effect of wavelength that is not sufficiently taken into account by the coefficient j v. Coefficient b is determined by Eq. (3); as for the quadratic damping approach the coefficients j o and K b have been adjusted (j oK b = kVh). The standard deviation between the modified coefficient b and the coefficients that have been derived from measurements is r b = 19.9 s2/m2 (93.0%, with a = 0.89 s/m). The wave decay inside the breakwater core (according to Eq. (13)) is plotted in Fig. 8c against the relative distance (a + bj vH 0) exp (0.5kaj vx) bj vH 0. The

wave heights in the most seaward part of the structure are slightly underestimated by the polynomial approach while the wave heights in the landward part of the structure are approximated very well. A direct comparison of measured and predicted wave heights is plotted in Fig. 8d. The calculated wave heights are on average 2.2% larger than the measured values and have a standard deviation r H = 0.026 m (16.8%). The largest wave heights (H N 0.4 m) in the most seaward part of the breakwater core are underestimated. If the experimentally determined linear and quadratic Forchheimer coefficients (a = 0.89 s/m and b = 22.9 s2/m2) are applied the wave heights inside the breakwater will be overestimated on average by 8.0%, the standard deviation is increased to 0.046 m (32.7%). The polynomial approach according to Eq. (13) (with an empirical adaptation of the quadratic resistance coefficient b) provides a good approximation of the actual wave height decay inside a breakwater. However, the accuracy of the results is limited by

722

M.O. Muttray, H. Oumeraci / Coastal Engineering 52 (2005) 709–725

the coefficient j v, which does not cover the effect of wavelength completely. 3.2.4. Extended polynomial approach In order to consider the effect of wavelength on the wave damping properly an empirical coefficient fix is added to the damping approach according to Eq. (13): a   H ð xÞ ¼
ð16Þ  a p þ bjv exp ajv jx xV  bjv H0 2 with: xV = x  x 0. For regular waves, the wave decay inside the breakwater core according to Eq. (16) is plotted in Fig. 9. The experimentally derived Forchheimer coefficients a = 0.89 s/m and b = 22.9 s2/m2 (see Table 4), an internal wave number kV (Eq. 15) and the following coefficient fix have been applied: ð17Þ jx ¼ 1:5kVh The standard deviation between Eq. (17) and coefficients fix that have been derived from measurements

is r j x = 0.385(34.3%) (Fig. 9a). The local wave heights inside the structure according to Eq. (16) are on average 0.6% larger than the measured wave heights, the standard deviation is r H = 0.021 m or 14.9% (Fig. 9b). From Fig. 9c it can be seen that the actual wave height decay inside a breakwater core is well described by the extended polynomial approach (Eq. (16)). For wave spectra, the decay of significant wave heights inside the breakwater is plotted in Fig. 10. The extended polynomial approach according to Eq. (16) has been applied. The experimentally derived Forchheimer coefficients a = 0.89 s/m and b = 22.9 s2/m2 and an internal wave number kV have been used as for regular waves. The coefficient j x has been slightly modified: jx ¼ 1:25kVh

The standard deviation between Eq. (18) and coefficients j x that have been derived from measurements is r j x = 0.226 or 25.9% (Fig. 10a). The calcu-

(a) Coefficient κ x

(b) Measurement & Prediction 0.8

σx = 0.385 (34.3%)

calculated H(x) [m]

coefficient κ x [-]

5 4 3 2

κ x = 1.5 kh

1

ð18Þ

σH = 0.021 (14.9%) µH = 1.006

0.6 0.4 0.2 0

0 0

0.4

0.8

1.2

1.6

2

0

0.2

relative water depth k’h [-]

0.4

0.6

0.8

measured H(x) [m]

(c) Damping Function

H(x)/H 0 [-]

1

T = 3s T = 4s T = 5s T = 6s T = 8s T = 10s

a = 0.89s/m b = 22.9s2/m2 κx = 1.5 k’h

0.8 0.6

damping function

0.4 0.2 0 0.8

1

2

4

6

8

10

(a + b κv H 0 ) exp(π/2 a κ v κ x x’) - b κ v H0 [s/m2 ]

20

Fig. 9. Extended polynomial wave damping approach for regular waves: (a) coefficient j x vs. relative water depth kVh, (b) comparison of measured and calculated local wave heights H(x) and (c) wave decay inside the breakwater core.

M.O. Muttray, H. Oumeraci / Coastal Engineering 52 (2005) 709–725

(a) Coefficient κx

(b) Measurement & Prediction calculated Hm0 (x) [m]

coefficient κx [-]

3

σx = 0.226 (25.9%)

2.5 2 1.5

κx = 1.25 kh

1 0.5 0 0

0.4

0.8

1.2

723

1.6

2

0.8

σH = 0.039 (29.5%) µH = 0.974

0.6 0.4 0.2 0 0

0.2

relative water depth k’h [-]

0.4

0.6

0.8

measured Hm0 (x) [m]

(c) Damping Function

Hm0 (x)/H0,m0 [-]

1

Tp = 3s Tp = 4s Tp = 5s Tp = 6s Tp = 8s Tp = 10s

a = 0.89s/m b = 22.9s2/m2 κx = 1.25 k’h

0.8

damping function

0.6 0.4 0.2 0 0.8

1

2

4

6

8

10

(a + b κ v H 0,m0 ) exp(π/2 a κ v κ x x’) - b κ v H 0,m0 [s/m2 ]

20

Fig. 10. Extended polynomial wave damping approach for wave spectra: (a) coefficient j x vs. relative water depth kVh, (b) comparison of measured and calculated local wave heights H m0(x) and (c) wave decay inside the breakwater core.

lated local wave heights inside the structure are on average 2.6% smaller than the measured wave heights, the standard deviation is r H = 0.039 m or 29.5% (Fig. 10b). The relative wave height inside the structure H m0(x) = H 0, m0 is plotted in Fig. 10c against the relative distance. The wave heights in the seaward part of the breakwater are slightly overestimated while the wave heights in the landward part are slightly underestimated. In some of the experiments with wave periods T p = 5, 6 and 8 s, the wave height measurements are distorted by binternal wave overtoppingQ (the wave run-up at the intersection between filter layer and core reaches the crest level of the breakwater core and generates an additional water surface for the wave gauges in this part of the breakwater). Thus it appears as if these waves were propagating a certain distance into the breakwater without significant damping. In all cases without binternal wave overtoppingQ the decay of significant wave heights H m0 inside the breakwater can be reasonably assessed by Eq. (16) for irregular waves.

4. Concluding remarks The wave damping inside a rubble mound breakwater has been studied theoretically and experimentally. The theoretical concept has been based on the assumption of a homogeneous core material. The wave kinematics have been derived from linear wave theory and the wavelength inside the structure has been determined from a linearised dispersion equation for porous media. An average discharge velocity has been considered (averaged over wave period and water depth). The hydraulic gradient has been approximated by the gradient of wave or pressure height. It has been further assumed that the actual hydraulic resistance can be approximated either by a linear, a quadratic or by a polynomial (linear and quadratic) resistance. The effect of non-stationary flow (drag force) has not been considered explicitly. Despite these assumptions and simplifications the damping functions that have been derived from above analysis reflect the governing physical processes better than available approaches such as Eq. (6).

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M.O. Muttray, H. Oumeraci / Coastal Engineering 52 (2005) 709–725

The actual wave damping can be well described by a linear damping model and the corresponding exponential decrease. The linear Forchheimer coefficient a is not appropriate and has been replaced by an equivalent linear damping coefficient a eq, which takes both laminar and turbulent losses into account and has to be determined iteratively. A quadratic damping model with a damping coefficient that corresponds to the quadratic Forchheimer coefficient b does not provide any significant improvement while a polynomial approach appears to be very promising. The polynomial approach is mainly advantageous due to the fact that Forchheimer coefficients a and b can be directly applied as damping coefficients in the polynomial model. It was found that the polynomial damping model underestimates the effect of wavelength. Thus, an empirical correction has been applied (extended polynomial approach, Eq. (16)) that compensates the shortcomings of the theoretical approach with respect to averaged discharge velocity and linearised dispersion equation. The extended polynomial damping model (with empirical corrections according to Eqs. (17) and (18)) is applicable for regular and irregular waves if the hydraulic resistance of the porous medium can be approximated by the Forchheimer equation. Thus, the new approach is not restricted to certain breakwater geometries or wave conditions. However, in the case of wave overtopping or binternal wave overtoppingQ (infiltration into the core from the breakwater crest) the pressure distribution inside the breakwater and the internal wave decay may differ significantly from the proposed model.

Acknowledgements The support of the German Research Foundation (DFG) within the Basic Research Unit SFB 205, project B13 (bDesign of rubble mound breakwatersQ) and within the research programme bDesign wave parameters for coastal structuresQ (Ou 1/3–1,2,3) is gratefully acknowledged. Further the essential contributions of Christian Pabst amongst many others in the experimental study, of Erik Winkel in the data analysis and of Dr. Deborah Wood in the theoretical study shall be mentioned.

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